Transcendence results on the generating functions of the characteristic functions of certain self-generating sets, II

Peter Bundschuh¹ ; Keijo Väänänen²

¹ Mathematisches Institut Universität zu Köln Weyertal 86-90 50931 Köln, Germany
² Department of Mathematical Sciences University of Oulu P.O. Box 3000 90014 Oulu, Finland

Acta Arithmetica, Tome 167 (2015) no. 3, pp. 239-249

Résumé

This article continues a previous paper by the authors. Here and there, the two power series $F(z)$ and $G(z)$, first introduced by Dilcher and Stolarsky and related to the so-called Stern polynomials, are studied analytically and arithmetically. More precisely, it is shown that the function field $\mathbb {C}(z)(F(z),F(z^4), G(z),G(z^4))$ has transcendence degree 3 over $\mathbb {C}(z)$. This main result contains the algebraic independence over $\mathbb {C}(z)$ of $G(z)$ and $G(z^4)$, as well as that of $F(z)$ and $F(z^4)$. The first statement is due to Adamczewski, whereas the second is our previous main result. Moreover, an arithmetical consequence of the transcendence degree claim is that, for any algebraic $\alpha $ with $0|\alpha |1$, the field $\mathbb {Q}(F(\alpha ),F(\alpha ^4),G(\alpha ),G(\alpha ^4))$ has transcendence degree 3 over $\mathbb {Q}$.

Détail

DOI : 10.4064/aa167-3-2

Keywords: article continues previous paper authors here there power series first introduced dilcher stolarsky related so called stern polynomials studied analytically arithmetically precisely shown function field mathbb has transcendence degree mathbb main result contains algebraic independence mathbb first statement due adamczewski whereas second previous main result moreover arithmetical consequence transcendence degree claim algebraic alpha alpha field mathbb alpha alpha alpha alpha has transcendence degree mathbb